L(s) = 1 | − 3-s + 9-s − 6·11-s + 13-s + 2·17-s + 4·23-s − 5·25-s − 27-s + 6·29-s − 4·31-s + 6·33-s + 2·37-s − 39-s − 4·43-s + 10·47-s − 7·49-s − 2·51-s + 10·53-s + 6·59-s + 6·61-s + 12·67-s − 4·69-s + 2·71-s + 6·73-s + 5·75-s − 16·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 0.485·17-s + 0.834·23-s − 25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 1.04·33-s + 0.328·37-s − 0.160·39-s − 0.609·43-s + 1.45·47-s − 49-s − 0.280·51-s + 1.37·53-s + 0.781·59-s + 0.768·61-s + 1.46·67-s − 0.481·69-s + 0.237·71-s + 0.702·73-s + 0.577·75-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.174217462\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174217462\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.840201352207270406224561644624, −8.067056822784774402058659889799, −7.42908510259424397801194980381, −6.61084979929999274945585321507, −5.56597126052820488051950415035, −5.25010915829523959096570683485, −4.21259172810034728687825987566, −3.12961052873891392238054519582, −2.16189239262552998925883882829, −0.69386451230752267708646070580,
0.69386451230752267708646070580, 2.16189239262552998925883882829, 3.12961052873891392238054519582, 4.21259172810034728687825987566, 5.25010915829523959096570683485, 5.56597126052820488051950415035, 6.61084979929999274945585321507, 7.42908510259424397801194980381, 8.067056822784774402058659889799, 8.840201352207270406224561644624